Logic and Proof
When detectives Benson and Stabler search a suspect's apartment, they aren't looking for just anything to give to the prosecutor. They need solid, tangible, and legally obtained evidence. (Although "legally obtained" is more of a suggestion than a requirement on Law and Order.) The point is that no jury or judge will convict based on a hunch or a guess. They'll need proof.
Think of proofs as arguments. If you are debating with your parents over your curfew and your main point is that you're responsible enough to stay out late, it's important to have reasons supporting this claim. Just an FYI, now wouldn't be a good time to bring up that flashing red light you ran a couple sections back.
The ability to support your statements with reasons is the essence of effective debate, and is probably the most important skill to gain from studying geometry (and mathematics in general). It will help you to be a more convincing speaker and might just get your curfew pushed back an hour or five.